The stability of spike solutions to the one-dimensional Gierer–Meinhardt model

Iron, David; Ward, Michael J.; Wei, Juncheng

doi:10.1016/s0167-2789(00)00206-2

Cited by 217 publications

(343 citation statements)

References 12 publications

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“…The stability of symmetric k-spike equilibrium solutions to (1.4) was analyzed in [17] for the case τ = 0 and in [46] for τ > 0. Asymmetric k-spike equilibria were constructed in [45] and [10], and a partial stability analysis for asymmetric patterns was given in [45].…”

Section: In the Intermediate Regime O(1)mentioning

confidence: 99%

“…However, the rigorous proofs given in [47] do not directly carry over to the one-dimensional situation for the GS model studied here. The stability of a one-spike solution to (1.4) on the infinite line was studied in [9], and the dynamics of spikes was studied in [16], [42], and [43]. For the GM model, the relationship between translational instabilities of symmetric k-spike patterns and the emergence of asymmetric spike patterns is emphasized in [45] and [47].…”

Section: In the Intermediate Regime O(1)mentioning

confidence: 99%

“…A problem similar to (3.25) was formally derived in Proposition 8 of [17] in the context of the GM model (1.4) with exponent set (p, q, m, s), where ζ ≡ qm/(p − 1) − (1 + s) > 0. By comparing Proposition 8 of [17] with Principal Result 3.1 above, we conclude that (3.25) for the GS model is equivalent to the corresponding problem for the small eigenvalues of a GM model with exponent set (p, q, m, s) = (2, s, 2, s), where s = (1 − U ± )/U ± .…”

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confidence: 96%

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Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Kolokolnikov

Ward

Wei

2006

Interfaces Free Bound.

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Slow translational instabilities of symmetric k-spike equilibria for the one-dimensional singularly perturbed two-component Gray-Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that k−1 small eigenvalues, governing the translational stability of a symmetric kspike pattern, simultaneously cross through zero at precisely the same parameter value at which k −1 different asymmetric k-spike equilibria bifurcate off of the symmetric k-spike equilibrium branch. These asymmetric equilibria have the general form SBB . . . BS (neglecting the positioning of the B and S spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter τ is asymptotically large as ε → 0. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory.

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Section: In the Intermediate Regime O(1)mentioning

confidence: 99%

Section: In the Intermediate Regime O(1)mentioning

confidence: 99%

mentioning

confidence: 96%

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Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Kolokolnikov

Ward

Wei

2006

Interfaces Free Bound.

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“…Starting with Turing's seminal paper [34], diffusion and cross-diffusion have been observed as causes of the spontaneous emergence of ordered structures, called patterns in a variety of non-equilibrium situations. They include the Gierer-Meinhardt model [35][36][37][38], the Sel'kov model [26,39], the Lotka-Volterra competition model [40][41][42] and the Lotka-Volterra predator-prey model [20,23,24,[43][44][45] and so on.…”

Section: Introductionmentioning

confidence: 99%

Qualitative analysis of a strongly coupled predator–prey system with modified Holling–Tanner functional response

Zeng

2018

Bound Value Probl

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“…Existence and stability of spiky steady states have been studied for 1-D in [13] and their instabilities have been investigated in [27]. For 2-D the existence and stability of multiple spikes has been investigated in [31], [32], [33].…”

Section: Introductionmentioning

confidence: 99%

Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients

Wei

Winter

2009

J Nonlinear Sci

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Abstract. We rigorously prove results on spiky patterns for the Gierer-Meinhardt system [10] with a jump discontinuity in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini andFirstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity.This localization principle for a spike is a new effect which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable.Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper.To derive these new effects in a mathematically rigorous way, we use analytical tools like LiapunovSchmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work [33].Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.1991 Mathematics Subject Classification. Primary 35B35, 76E30; Secondary 35B40, 76E06.

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The stability of spike solutions to the one-dimensional Gierer–Meinhardt model

Cited by 217 publications

References 12 publications

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Qualitative analysis of a strongly coupled predator–prey system with modified Holling–Tanner functional response

Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients

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